Efficient Bayesian Estimation and Uncertainty Quantification in Differential Equation Models

• Activity Number: 305
• Type: Topic Contributed
• Date/Time: Tuesday, August 11, 2015 : 8:30 AM to 10:20 AM
• Sponsor: Section on Bayesian Statistical Science
• Author(s): Prithwish Bhaumik* and Subhashis Ghosal
• Companies: North Carolina State University and North Carolina State University
• Keywords: Ordinary differential equation ; Runge-Kutta method ; approximate likelihood ; Bayesian inference ; spline smoothing ; Bernstein-von Mises theorem

Abstract

Ordinary differential equations (ODEs), used to model dynamic systems, contain unknown parameters which have to be estimated from the data. In the absence of analytical solution, one approach is to use four stage Runge-Kutta (RK4) method to solve the system numerically. The approximate solution is then used to construct an approximate likelihood. We assign a prior on the parameters and then draw posterior samples, but this method may be computationally expensive. There is a computationally efficient two-step Bayesian approach, where a posterior is induced on the parameters using a random series based on the B-spline basis functions. But the Bayes estimator obtained in this approach is not asymptotically efficient. In this paper we also suggest a modification of the two-step method by directly considering the distance between the function in the nonparametric model and that obtained from RK4 method. We establish a Bernstein-von Mises theorem for the posterior distribution of the parameter in both RK4 approximate likelihood based and modified two-step approaches. Unlike in the original two-step procedure, the precision matrix matches with the Fisher information matrix.